• An integer is a factor of another number if it divides the other number with remainder zero.
  • A prime number is an integer which has only two factors, 1 and itself.
  • A composite number is an integer which has more than two factors.
  • A prime factor of a number is a prime number which divides the number with 0 remainder
  • A composite number written as a product of prime numbers is called the prime factorization of the number.
Example:
54 = 2 x 3 x 3 x 3.

How do you find the prime factorization of a number?

The method of finding the prime factorization of a number consists of doing divisibility tests or trial divisions. The division is continued with successive quotients by prime divisors. The division is not possible if a division gives a prime quotient. Hence the successive divisions are continued till a prime quotient is obtained. The product of the prime divisors used in successive divisions and the final prime quotient is the prime factorization of the given number.

The process of division done at each level can be shown as branches in a factor tree, one branch showing the prime divisor and the other branch leading to the quotient obtained on division.

Example for finding the prime factorization of a number.

Find the prime factorization of 96.
Since 96 is even number, it is divisible by 2. Hence the first division of 96 by 2 gives the factors as
96 = 2 x 48.
The first branch of the factor tree will look like,
                                                                       96
                                                                        /\
                                                                       /  \
                                     Prime factor   →        2   48
The quotient 48 is again divisible by 2 and 48 = 2 x 24. The results of successive divisions by prime factors can be written as
24 = 2 x 12
12 = 2 x 6
6  = 2 x 3

Quotient 3 is a prime number and hence the division cannot be continued.  The entire process can be shown as a prime factorization tree as follows:
                                                                       96
                                                                        /\
                                                                       /  \
                                     Prime factor   →         48
                                                                          /\
                                                                         /  \
                                     Prime factor  →            24
                                                                             /\
                                                                            /  \
                                     Prime factor  →               12
                                                                                /\
                                                                               /  \
                                     Prime factor  →                  2    6
                                                                                   /\
                                                                                  /  \
                                     Prime factor  →                     2    3    ←  Prime factor.

The Prime factorization of 96 = 2 x 2 x 2 x 2 x 2 x 3      or using exponents  96 = 25 x 3.